Distributionally Safe Reinforcement Learning under Model Uncertainty: A Single-Level Approach by Differentiable Convex Programming

Safety assurance is uncompromisable for safety-critical environments with the presence of drastic model uncertainties (e.g., distributional shift), especially with humans in the loop. However, incorporating uncertainty in safe learning will naturally lead to a bi-level problem, where at the lower level the (worst-case) safety constraint is evaluated within the uncertainty ambiguity set. In this paper, we present a tractable distributionally safe reinforcement learning framework to enforce safety under a distributional shift measured by a Wasserstein metric. To improve the tractability, we first use duality theory to transform the lower-level optimization from infinite-dimensional probability space where distributional shift is measured, to a finite-dimensional parametric space. Moreover, by differentiable convex programming, the bi-level safe learning problem is further reduced to a single-level one with two sequential computationally efficient modules: a convex quadratic program to guarantee safety followed by a projected gradient ascent to simultaneously find the worst-case uncertainty. This end-to-end differentiable framework with safety constraints, to the best of our knowledge, is the first tractable single-level solution to address distributional safety. We test our approach on first and second-order systems with varying complexities and compare our results with the uncertainty-agnostic policies, where our approach demonstrates a significant improvement on safety guarantees.

Wasserstein Distributionally Robust Control Barrier Function using Conditional Value-at-Risk with Differentiable Convex Programming

Control Barrier functions (CBFs) have attracted extensive attention for designing safe controllers for their deployment in real-world safety-critical systems. However, the perception of the surrounding environment is often subject to stochasticity and further distributional shift from the nominal one. In this paper, we present distributional robust CBF (DR-CBF) to achieve resilience under distributional shift while keeping the advantages of CBF, such as computational efficacy and forward invariance. To achieve this goal, we first propose a single-level convex reformulation to estimate the conditional value at risk (CVaR) of the safety constraints under distributional shift measured by a Wasserstein metric, which is by nature tri-level programming. Moreover, to construct a control barrier condition to enforce the forward invariance of the CVaR, the technique of differentiable convex programming is applied to enable differentiation through the optimization layer of CVaR estimation. We also provide an approximate variant of DR-CBF for higher-order systems. Simulation results are presented to validate the chance-constrained safety guarantee under the distributional shift in both first and second-order systems.

On the Optimality, Stability, and Feasibility of Control Barrier Functions: An Adaptive Learning-Based Approach

Safety has been a critical issue for the deployment of learning-based approaches in real-world applications. To address this issue, control barrier function (CBF) and its variants have attracted extensive attention for safety-critical control. However, due to the myopic one-step nature of CBF and the lack of principled methods to design the class-$\mathcal{K}$ functions, there are still fundamental limitations of current CBFs: optimality, stability, and feasibility. In this paper, we proposed a novel and unified approach to address these limitations with Adaptive Multi-step Control Barrier Function (AM-CBF), where we parameterize the class-$\mathcal{K}$ function by a neural network and train it together with the reinforcement learning policy. Moreover, to mitigate the myopic nature, we propose a novel \textit{multi-step training and single-step execution} paradigm to make CBF farsighted while the execution remains solving a single-step convex quadratic program. Our method is evaluated on the first and second-order systems in various scenarios, where our approach outperforms the conventional CBF both qualitatively and quantitatively.